I'm thinking about contributing to the ArchWiki by means of translating English articles into French. This would allow me to kill two birds with one stone as I will be contributing to the community and improving and practising my French.

But how does this process work? Can I just create an account on the wiki and start editing? Or do I need to go through some approval process? Any advice would be greatly appreciated

Create an account and get going, pretty much. I tend to watch the pages that I make modifications to, and be sure to check the "minor edit" box when you're just fixing a spelling mistake or something like that. I'm not familiar enough with the wiki though to know if it notifies you by default of new Talk/Discussion items on articles that you're watching.